Cutting a "channel" halfway into a shape?
You can use knife project.

By hiding the faces you don't want to cut, you can control which faces get cut with the shape and which are left uncut.

Example:

With the shape object selected,  Shift-Select the object you want to cut and enter edit mode:



Select the bottom faces (the ones you want to preserve with no cuts) and Hide them (H)



Then in top orthogonal view press the Space Bar and type Knife Project. Enable Cut Through** and then unhide the hidden faces (AltH)